The fermionic seminorm, as defined in (REF ) by the maximum transition amplitude within the \(\eta \) -electron manifold, provides a reasonable metric for quantifying the error of digital quantum simulation with initial-state constraints. Indeed, a seminorm similar to our definition was used by Somma [1]} for analyzing quantum simulation of bosonic Hamiltonians. However, we point out that this is not the only error metric that takes the prior knowledge of the initial state into account. Recent work [2]} analyzed the low-energy simulation of \(k\) -local frustration-free Hamiltonians by computing the spectral norm of Trotter error projected on the low-energy subspace. However, the following proposition shows that these two error metrics are the same for fermionic systems.
